1. Field of the Invention
The present invention relates to methods and electronic apparatus for processing signals to produce high resolution signal information from an input signal consisting of a partial wave function. Such processing is useful, for example, in transmitting data across a transmission line at many times conventional bit rates, as in a modem network. Other applications include obtaining improved mechanical efficiencies in optical instrumentation and improved signal recording density on recording mediums.
2. Description of the Prior Art
In the signal processing art, certain advantages can be obtained by converting signals from a Z-transform state to the inverse Z-transform state, and vice versa. Moreover, some signals, such as that produced by a Michelson Interferometer spectrometer, are originated in a high resolution Z-transform state. Such a signal must be converted to a high resolution spectral signal by an inverse Z-transformer device in order to be useful or meaningful.
For example, in the Michelson Interferometer spectrometer, the usable output signal is the spectrum of individual light waves identifiable one from the other in amplitude and position along the wavenumber or wavelength axis.
As used in this specification, Z-transform is treated as the generic mathematical operation, and the discrete Fourier transform is used as a specific exemplary mathematical operation for convenience. When reference is made herein to discrete Fourier transformation, it is to be understood that the invention is not to be considered as limited to Fourier or any other type of transformation system.
When the Z-transform is truncated, a degraded data signal results, because the inverse Z-transform yields a low-resolution partial wave spectrum. The truncated Z-transform signal is thus the mathematical equivalent of a convolution of the high-resolution spectral signal and a Z-transformed window function (e.g. sync or Gaussian function). The problem to be solved then is to deconvolve these two functions (i.e. effect restoration of the degrated data signal by removing the effects of the truncating signal component) to reconstruct the desired high resolution signal. Deconvolution refers to the process of deriving an estimate of the input of a system from its output knowledge. To recover the input, it is necessary to solve a convolution equation. this is possible to do only when the system impulse response is known.
The deconvolution process to reconstruct the information signal represented by a truncated Z-transformed signal has been investigated by Remy Prost and Robert Goutte, as reported in their articles "Deconvolution When The Convolution Kernel Has No Inverse", IEEE, Volume ASSP-25, No. 6 (1977), Pages 542-549, and "Non-Iterative and N-Steps Iterative Support-Constrained Deconvolution Algorithmns", Signal Processing II: Theories and Application, Elsevier Science Publishers B.V. (North-Holland) (1983), Pages 459-498. Both of the aforementioned references are incorporated herein by reference. These references treat the restoration of degraded data involving a convolution kernel that has no inverse, i.e. the system has an impulse response whose Fourier transform has a cutoff frequency.
These prior art systems are based on the fundamental requirement that a priori information is essential in analyzing a convolution kernel that has no inverse. The earlier Prost and Goutte publication discusses an iterative deconvolution algorithm, and their later publication teaches non-iterative and N-steps deconvolution algorithms. The a priori knowledge requirement is common to all disclosed deconvolution algorithms, and is defined in the earlier Prost and Goutte publication as the knowledge of an interval which includes the support of the signal to be restored, and is a necessary and sufficient constraint required for the mathematical resolution of the deconvolution problem.
Thus, when processing a truncated (or frequency cutoff) Z-transform signal, known deconvolution methods can be incorporated to produce a reconstructed version of the signal of interest in the system, i.e. what prost and Goutte refer to as the system input.
Such prior art systems are adequate for processing continuous or purely analog versions of the truncated Z-transform signal. However, when the truncated Z-transform signal is converted to digital form for further processing, serious further system inaccuracies are introduced.
The ordinary system clock available to digitize an analog signal for further processing, if used to sample a continuous non- truncated Z-transform signal, is also used as the timing source to perform the inverse Z-transform signal, and the identical sampling rates used in the digital domain permit easy and accurate Z-transformations because of the presence of an exact sampled value occurring at each clock time. In an encoding-transmission-decoding system (e.g. modem network), the recreated clock at the receiving end, from self-clocking techniques for example, is relatively easy to produce from the digital data extant in the transmission channel. Again, a one-for-one relationship between the sampled and transmitted values of data and the derived clock precludes the loss of any transmitted information.
A formidable problem results, however, when the Z-transform signal to be digitized and transmitted is truncated. The system clock used in the digitizing process results in fewer sampling points of the Z-transform being transmitted because of the truncation. While an inverse Z-transform can be produced when the digitized (and further processed) truncated Z-transform signal is converted back to its analog equivalent at the receiver, the available number of sample points to be analog converted is the same as that sampled at the analog-to-digital converter. As a consequence, the envelope of the recovered analog equivalent of the truncated Z-transform signal is devoid of distribution points necessary to accurately reconstruct the input signal by the aforedescribed deconvolution techniques.
Stated another way, in a continuous and purely analog system, the high resolution deconvolution techniques depend upon the existence of accurate information in that interval of the signal to be restored between the boundaries (support) of the truncated Z-transform. Accordingly, the bounded interval must be mathematically continuous or at least must have the sufficient distribution components needed in carrying out the deconvolution algorithm. In a system operating on a Z-transform resulting from a digital to analog conversion, this would be the equivalent of using a system clock of infinite frequency (for the continuous analog) or of a frequency compatible with the resolution constraints of the deconvolver (for the "sufficient components" analogy). Clock generators of prior art systems are not capable of providing the sampling rates necessary to meet these requirements. Moreover, as will become evident later in this description, merely increasing the sampling rate by raising the frequency of the sampling clock is not the total solution to the problem, because limited bandwidth devices and transmission channels place practical constraints on sampling rate. Without frequency coherence and (in a transmit/receive system) without accurate recreation of a high frequency sampling clock from the transmitted data, a distorted and noisy inverse Z-transform will result, and deconvolution will yield an inadequate or useless system output signal.
Accordingly, there is a need in the art for a signal processing system that will permit retrieval of a high resolution output from the deconvolution of the inverse transform, in the digital domain, of a truncated Z-transform signal, with output signal resolution comparable to non-truncated, purely analog systems. The super resolving partial wave signal processing system of the present invention satisfies that need.
To illustrate the need in the art for such a system, this specification supports two practical examples incorporating principles of the invention; a super-resolving digital modulator/demodulator (modem), and a superresolving Fourier transform spectrometer.
In the modem example, the present invention relates to electronic apparatus for transmitting and receiving digital data over a signal transmission channel. Since the telephone lines used for transmitting modem information are in fact bandwidth limited analog signal channels, the modem transmission channel will be referred to herein as an analog signal channel, although the data to be transmitted in a typical modem network is carried in the alterations of a carrier signal, thereby taking on a digital nature.
The familiar dial-up modem which connects a personal computer to the telephone system allows the computer to transmit and receive relatively low data rate information to and from other units equipped with modems, thereby forming a modem network. Modems capable of relatively higher data rates are used to connect large mainframe computers through microwave and satellite links. High data rate modems are also used to transmit and receive digitized speech and facsimile information over special-purpose leased telephone lines. These are only a few of an extremely large number of applications for modem-type devices.
Present-day modems typically use frequency or phase modulation of a fixed-frequency carrier signal. This is a simple technique producing a modulated signal bandwidth in hertz approximately equal to the number of bits transmitted per second (quadriphase modulation).
Dial-up modems typically use this method to achieve data rate of up to about 2400 bits/sec over standard telephone lines.
Although adequate for many applications, this data rate capability is unacceptably low for applications such as digitized voice, facsimile, and other high density data transmission applications. For example, for a 1k.times.1k bi-tone image, one million bits must be transmitted, taking about seven minutes at 2400 b/s. Almost all communications systems using analog channels would greatly benefit by the use of a modem able to transmit and receive at higher bit rates over existing channels. The super-resolving Z-transform signal processing system of the present invention satisfies this need.
As related to Fourier transform spectrometers, the super-resolving Z-transform signal processing system of the present invention relates to electro-optical apparatus for obtaining the optical spectrum of infrared or visible light.
Analysis of optical spectra is necessary in a variety of applications. For example, in order to determine the composition of a chemical sample, infrared light from a glowing wire is focused onto the sample, and its transmission spectrum is measured. The transmitted infrared spectrum constitutes a complex electronic "fingerprint" and is determined by the various molecular absorption characteristics of various organic and inorganic compounds in the sample. The infrared spectrum of light radiated from various objects can likewise be used to determine the chemical constituents of the irradiating source (cf emission spectroscopy). Visible and infrared spectroscopy has contributed to many discoveries in the field of organic chemistry, astronomy, plasma physics, and other sciences.
Optical spectra can be obtained by known methods using simple means such as gratings or prisms (cf dispersive spectrometers), but such methods are not very sensitive, and are extremely time consuming. A major advance in this art was the introduction of the Fourier transform spectrometer (FTS) which essentially consists of a Michelson interferometer whose output signal (the interferogram) when mathematically transformed, by inverse Fourier transformation, constitutes the optical spectrum. The FTS has two principal advantages over grating instruments:
(a) All frequencies are analyzed simultaneously, so the optical spectrum is obtained very quickly (Fellget's advantage); and
(b) The entire optical aperture is used, not just a slit, so the FTS is more sensitive (Jacquinot's advantage).
The Fourier transform infrared (FTIR) spectrometer has been developed into several commercial instruments, and represents an important tool for modern physicists and chemists.
For their advantages, Fourier transform spectrometers have one principal disadvantage which results in their being bulky, delicate, and expensive. This disadvantage lies in the fundamental fact that the optical resolution is inversely proportional to the distance that a mirror is moved in one arm of the interferometer. Therefore, in order to obtain a typical 1 wave number (inverse centimeter) spectrum, the mirror must be moved about 1 centimeter. in order to achieve acceptable optical signal-to-noise ratios, the mirror must not tilt appreciably (to the order of an optical wavelength, about 1 micrometer) over the translational movement of 1 centimeter. Furthermore, the mirror motion must be approximately linear with time. In order to achieve these three requirements, expensive machine parts, air bearings, and complex electronic servo-mechanisms must be employed. Such components are delicate, expensive, bulky, and difficult to keep aligned. Present units work reliably only in the laboratory, are not portable, and will not operate in any position except horizontally.
It is therefore clear that there is a need in the art for an electro-optical servo-mechanism which when used with a Michelson interferometer requires only a small amount of mechanical movement of the operative element (about 100 times less than present instruments). The super-resolving partial wave signal processing system of the present invention fulfills this need in the art.